91Ó°ÊÓ

The equal sides of an isosceles triangles are each 8 in. long, the base being variable. Show that the triangle of maximum area is the one which has a right angle. Take one of the base angles as the independent variable, \(\phi\).

A gutter is to be made out of a long strip of copper 9 in. wide by bending the strip along two lines parallel to the edges and distant respectively 3 in. from an edge. Thus the cross-section will be a broken line, made up of three straight lines, each 3 in. long. How wide should the gutter be at the top, in order that its carrying capacity may be as great as possible?

At what points of the cardioid is the tangent perpendicular to the axis of the curve?

A statue ten feet high stands on a pedestal that is \(50 \mathrm{ft}\). high. How far ought a man whose eyes are 5 ft. above the ground to stand from the pedestal in order that the statue may subtend the greatest possible angle?

Plot the curve, \(\quad r=a \cos 3 \theta\), taking \(a=5 \mathrm{~cm} .\) Show that $$ \cot \psi=-3 \tan 3 \theta. $$

The captain of a man-of-war saw, one dark night, a privateersman crossing his path at right angles and at a distance ahead of \(c\) miles. The privateersman was making \(a\) miles an hour, while the man-of-war could make only \(b\) miles in the same time. The captain's only hope was to cross the track of the privateersman at as short a distance as possible under his stern, and to disable him by one or two well-directed shots; so the ship's lights were put out and her course altered in accordance with this plan. Show that the man-of- war crossed the privateersman's track \(\frac{c}{b} \sqrt{a^{2}-b^{2}}\) miles astern of the latter. If \(a=b\), this result is absurd. Explain.

Into a full conical wine-glass whose depth is \(\alpha\) and generating angle \(\alpha\) there is carefully dropped a spherical ball of such a size as to cause the greatest overflow. Show that the radius of the ball is $$ \frac{a \sin \alpha}{\sin \alpha+\cos 2 \alpha} $$

The versed sine and the coversed sine are defined as follows : $$\text { vers } x=1-\cos x ; \quad \text { covers } x=1-\sin x$$ $$ u=\frac{\cos \phi}{\sqrt{1-k^{2} \sin ^{2} \phi}} $$

The versed sine and the coversed sine are defined as follows : $$\text { vers } x=1-\cos x ; \quad \text { covers } x=1-\sin x$$ $$ x \cos y=\sin (x+y) \cdot \quad \frac{d y}{d x}=-\frac{\cos (x+y)-\cos y}{\sin (x+y)+x \sin y} $$

The versed sine and the coversed sine are defined as follows : $$\text { vers } x=1-\cos x ; \quad \text { covers } x=1-\sin x$$ $$ \tan \theta+\tan \phi=2 \tan \phi \tan \theta $$